Some Classes Of Constrained Matrix Equations Problem

The constrained matrix equation problem is to find the solution of a matrix equation in a constrained matrix set.Different constrained conditions and different matrix equations lead to unlike costrained matrix equation problems. Constrained matrix equation problem is widelyused in many fields such as structure design, structure dynamics, system identifiation, automatics control theory, vibration theory. The theory of constrained matrix quation problem is just prompted to develop quickly by many different question arisen in those fields, and it has become one of the topics of very active research in the cmputational mathematics.The main problems of this paper ar described as follows:Prodlem I Given X_o∈C_p×q,find X∈S_E such that where S_E is a given matrix set. Prodlem II Given X^*∈C^p×q,find X∈S_E such that where S_E is the solution set of Problem I.The main works of this paper are as follows:1. The problems of finding the least squares solutions, reflexive solutions,anti-reflexive solutions of the matrix equation (AX,XB) = (C,D) with a submatrix constraint are respectively studied. By using the Projection Theorem, we cantransform the least squares problem of the matrix equation into the equation problem of the matrix equation. Thus, the difficulty that the invariance of the Frobenius normonly holds for orthogonal matrices, but does not hold for nonsingular matrices can besmoothed away. The general expressions of Problem I are obtained respectively. In addition, the optimal approximation solutions of Problem II are derived respectively.2. The problems of finding symmetric solutions, anti-symmetric solutions of the matrix equation (AXA^T ,BXB^T,AXB^T)=(E,F,G) with a submatrix constraint are respectively studied. By making use of the special properties of the matrix equation, we change Problems I, II of matrix equation (AXA^T,BXB^T,AXB^T)- (E,F,G) with a submatrix constraint into Problems I, II of matrix equation AXA^T = B with a submatrix constraint.